NON-INTEGRABILITY OF SOME PROBLEMS IN CELESTIAL
MECHANICS Sergi Simon i Estrada
Departament de Matem`atica Aplicada i An`alisi
Universitat de Barcelona
Mem`oria presentada per a aspirar al grau de
Doctor en Matem`atiques per la Universitat de Barcelona.
Certifiquem que la present mem`oria ha estat realitzada per Sergi Simon i Estrada
i codirigida per nosaltres.
Barcelona, 14 de maig de 2007,
Juan J. Morales-Ruiz Carles Sim´o i Torres
i a l’Ainhoa
First of all, my special words of gratitude must go to my PhD advisors, Juan J. Morales-Ruiz and Carles Sim´o, for the path jointly traveled during the past years. I wish to thank Juan for introducing me head-first into the realm of differ- ential Galois theory, arguably one of the most elegant and hauntingly beautiful domains of Mathematics – and yet, perhaps surprisingly, a bluntly powerful one for practical purposes considering the incipient theory he himself created along with Jean-Pierre Ramis not long ago. I’m also obliged to Juan for his relentless patience, his continuous encouragement, his overtly instructive demeanor and his selfless and dedicated efforts at teaching me the basics (and the not-so-basics) of the Galoisian approach to the study of differential systems ever since I was an absolute beginner in the field. I wish to thank Carles for introducing me to Juan in the first place once I expressed my interests on integrability, for providing me with ambitious open problems to work with (and whose solution makes up for the bulk of this text), for initiating me in the deep territory of differential equations in general and Hamiltonians in particular, for conferring me a long-lasting affec- tion, and interest, for Celestial Mechanics, and, it should be said, for showing me that a certain degree of bold temerity and a taste for mathematical “tours de force” isn’t that bad, after all. Thank you both, for teaching me the things which don’t come up in the books and, most importantly, for teaching me how to gather them myself.
Concerning my six-year stay as a member of the Departament de Matem`atica Aplicada i An`alisi of the Universitat de Barcelona, a first and special mention must be made to my bureau partner for the whole time span, namely Salvador Rodr´ıguez, coincidentally also my classmate all through the previous degree stud- ies; it is a pleasure and an honor to have you as a deskmate and as a friend. A scenery appears right behind in which some faces stand out for posterity, espe- cially those of Eva Carpio, Ariadna Farr´es, Manuel Marcote, Estrella Olmedo and Arturo Vieiro, to say a few; it has also been a pleasure to share the everyday treadmill with you folks. And I certainly wouldn’t like Nati Civil believing I’ve forgotten her; her combination of kind demeanor and extreme effectiveness would be much of a stretch to forget. I must also mention Primitivo Acosta-Hum´anez and David Bl´azquez-Sanz, with whom I’ve shared good moments and with whom I’ve started an ambitious, and hopefully successful, mathematical agenda.
Prior to the conclusion of the present dissertation, the kind invitation by professor Jean-Pierre Ramis made it possible for me to spend three-and-a half months at the Universit´e Paul Sabatier in Toulouse (France), in view of fulfilling one of the prerequisites for the European doctorate certificate. This is already one
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collaboration (and articles) with Jean-Pierre, Carles, Juan, Olivier Pujol, Jos´e- Philippe P´erez, and Jacques-Arthur Weil from Limoges. It furthermore focused my interest on higher variational equations, which are likely to play a significant role in my ensuing research. My stay in Toulouse was also deeply enriching on other levels, thanks to the hospitality shown by people of the likes of Mathieu Anel, Benjamin Audoux, Aur´elie Cavaille, Yohann Genzmer (and Johanna), Anne Granier, Philippe Lohrmann, C´ecile Poirier, Nicolas Puignau, Maxime Rebout, Julien Roques, Gitta Sabiini or Landry Salle, among others.
I am also deeply indebted to Jacques-Arthur Weil for his dedicated efforts in providing me with a post-doctoral position at the Universit´e de Limoges; the fact these efforts have come up as successful couldn’t be better news, and at this point I must say I also thank Jean-Pierre Ramis’ intervention in endorsing my application. On a mathematical level, my interaction, if sporadic, has also been satisfactory with a series of fellow mathematicians from abroad; among them, I wish to thank useful comments by Alain Albouy, Andrzej Maciejewski, Maria Przybylska, and Alexei Tsygvintsev, among others.
A very special mention must be made of my family: I would like to thank my mother M. `Angels and sister Nhoa for their constant backing, since they have seen most of the backstage of the efforts this thesis has taken. The fact that research in Mathematics is usually a series of ups and downs is no surprise but they have seen most of the ups and downs first-hand. I’ve felt supported and helped by both of you in every single way and I’m so very proud of you both, but well... you already know that, don’t you?
And I want to thank Ainhoa for the moments we have shared and for the bright perspectives ahead of us. These, along with your utmost patience, your unabashed support and your deep faith in me, are the real driving forces behind the conclusion of this thesis. Yet again, I’m not saying anything new, am I?
Barcelona, May 14th, 2007
Acknowledgements i
1 Introduction 1
1.1 Integrability of differential systems . . . 1
1.2 Historical note . . . 2
1.3 Original results . . . 4
1.3.1 Homogeneous potentials and N-Body Problems . . . 5
1.3.2 Non-integrability of Hill’s Problem . . . 6
1.4 General structure, notation and conventions . . . 6
2 Theoretical background 9 2.1 Useful results from Algebraic Geometry . . . 9
2.1.1 Preliminaries . . . 9
2.1.2 Linear algebraic groups and Lie algebras . . . 10
2.1.3 Rational invariants . . . 12
2.2 Notions of integrability . . . 15
2.2.1 Integrability of Hamiltonian systems . . . 15
2.2.2 Integrability of linear differential systems . . . 18
2.3 Morales-Ramis theory . . . 21
2.3.1 The general theory . . . 22
2.3.2 Special Morales-Ramis theory: homogeneous potentials . . 23
2.4 Basics in Celestial Mechanics . . . 27
2.4.1 The N-Body Problem . . . 27
2.4.2 Hill’s Lunar Problem . . . 35
3 The meromorphic non-integrability of some N-Body Problems 37 3.1 Introduction . . . 37
3.2 Preliminaries . . . 38
3.2.1 Statement of the main results . . . 38
3.2.2 Setup for the proof . . . 40
3.3 Proofs of Theorems 3.2.2 and 3.2.3 . . . 44
3.3.1 Proof of Theorem 3.2.2 . . . 44
3.3.2 Proof of Theorem 3.2.3 . . . 46
3.3.3 Proof isolate: N = 2m equal masses . . . 51 iii
4 The meromorphical non-integrability of Hill’s Lunar problem 55
4.1 Introduction . . . 55
4.1.1 Statement of the main results . . . 56
4.2 Proof of Lemma 4.1.1 . . . 57
4.2.1 Change of variables . . . 57
4.2.2 Solution of the new equation . . . 57
4.2.3 Singularities of φ2(t) . . . 58
4.3 Proof of Lemma 4.1.2 . . . 58
4.3.1 Layout of the system . . . 58
4.3.2 Fundamental matrix of the variational equations . . . 59
4.3.3 Relevant facts concerning Ψ(t) . . . 60
4.4 Proof of Theorem 4.1.3 . . . 62
4.5 Concluding statements . . . 69
5 Conclusions and work in progress 71 5.1 Overview . . . 71
5.2 Perspectives on Conjectures 5.1.1 and 5.1.2 . . . 72
5.2.1 The N-body problem with arbitrary masses . . . 72
5.2.2 Candidates for a partial result . . . 76
5.3 Hamiltonians with a homogeneous potential . . . 77
5.3.1 Higher variational equations . . . 77
Appendices 79 A Computations for Theorem 3.2.2 79 B Resum 83 B.1 Introducci´o . . . 83
B.1.1 Dues nocions d’integrabilitat en sistemes din`amics . . . 84
B.1.2 Alguns problemes de la Mec`anica Celeste . . . 87
B.2 Resultats originals . . . 89
B.2.1 Exist`encia d’una integral primera addicional . . . 90
B.2.2 Problemes de N Cossos . . . 91
B.2.3 La no-integrabilitat del Problema de Hill . . . 93
B.3 Agra¨ıments . . . 94
Introduction
1.1 Integrability of differential systems
The idea underlying any apprehension of an integrable dynamical system is the ability to make global assertions on the system’s evolution with respect to time.
Although the outcome of such assertions, usually called a solution, is fairly easy to characterize, giving the assertions themselves a strict definition has always proven a troublesome task, since each field of study has a specialized notion of
“solvability” of its own, seldom equivalent to the others’. The very concept of understanding a dynamical system is already difficult to define since the near- totality of cases will end in a non-trespassed threshold: to wit, the knowledge of a solution in closed form. Plainly speaking: there would be no controversy whatsoever on what integrability means (and hardly any need, by the way, to use such a word as chaos in ordinary differential systems) were the general solution of any dynamical system possible to find semi-algorithmically in a finite number of steps – a task nowadays unfeasible. There are attempts at partially circum- venting the latter obstacle, most notably the geometrical, also called qualitative, theory of differential equations (see [104], [109], [110], [130]) and perhaps most im- portantly the numerical simulation of solutions of differential equations based on qualitative theoretical results (see [123], [125]), and the computer-assisted proofs these simulations provide for (see [60], [74], [92], [124], [151], [161]), as well as the so-called algorithmic modeling paradigm relying on producing models from exper- imental data ([1]). However, what remains in all cases is an absolute dependence on disciplines (numerics, statistics, even algorithmic geometry) whose domain of application is peripheral to the theoretical groundwork, except when applied by researchers who conceive Mathematics as a science in and of its own rather than as a mere tool.
Thus naturally appears the phenomenon of specialization, so clearly visible in Section 2.2 and not as much a subterfuge as it may be an asset; it is our contention that most of the definitions of and conditions for integrability and non-integrability, including the ones explained in this text, are all their own part of more ambitious endeavors aimed precisely at integrating systems, at least those of a certain kind. Such an aim shows up most blatantly in the unrelenting effort at classifying all obstructions to “integrability” of dynamical systems, for instance the presence of certain special functions in their general solution. It
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may be argued, from a more general perspective, that this is not the shortest path to attain our final goal, but such a perspective is currently not available and there is one leitmotif underlying this outlook which already justifies the whole process wherever it may lead: the act of characterizing first integrals, and more generally systems for which these are easy to find, as anomalies in a wider, much more intricate context. In other words: foreshadowing the computation of exact solutions as a predictable accident in the hope of being able to predict it. Even if this is nothing but an act of self-delusion (as is any model, for that matter), the inference that it may work if done in a number of proper ways is currently enough for us.
1.2 Historical note
Arguably the cornerstone of Celestial Mechanics since it originated in Newton’s Principia, the N-Body Problem has long been seen in Astrophysics and Ap- plied Mathematics as an epitome of chaotic behavior; such behavior is retained in a significant amount in every model arising from it, especially by means of simplification. As a matter of fact, as we will recall below, most of the advances made in Applied Mathematics are precisely due to the presence of chaos in me- chanical systems directly or indirectly related to many-problems. However, there was a time during which both na¨ıvet´e and maximalism led philosophers to think otherwise. In keeping with this spirit, P.-S. de Laplace wrote, in 1814, the most famous paradigm in causal determinism:
“We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.”
([71], [72])
In this respect, the basis of modern science is firmly rooted on denial: if such an “intellect”, popularly referred to as Laplace’s demon, were to exist, it would most probably be beyond or outside the Universe, as well as independent and non-binding with respect to it (all of which contradicts Materialism), all present and past states would be knowledgeable (as opposed to Quantum Mechanics and Relativity), the concepts of irreversibility and entropy would be superfluous (hence, virtually obliterating the Second Law of Thermodynamics) and, above all, it would be possible to know all of the laws governing the Universe – a premise against which every single model in the History of Science, from “Panta Rhei”
to modern String Theory, may be directed at will if deemed pertinent. Even during Laplace’s lifetime, efforts as consistently intelligent as those made by C.
F. Gauss were bent strictly on questions of a pragmatic, specific and conceptually subservient sort, such as numerically solving Kepler’s equation derived from the
two-body problem in [42], rather than engaging in further exercises in futility such as the above quote.
The state of affairs by the mid-twentieth century was thus fairly predictable from the outset, especially for those accepting Science as an endless sediment of partial results assembled in an asymptotical quest for further open questions:
Classical Mechanics, as foreseen by Laplace and Newton, were seen by modern physicists as a barren land, a sterile discipline relegated to scholasticism. And yet, advances in Computer Science from the sixties onward, coupled with the insight of a number of researchers, brought about a series of theoretical results in turn show- ing new light on nonlinear dynamics; these results produced a slight abatement of the ostracism played on Classical Mechanics despite having their roots firmly planted on the perturbation theory by Poincar´e and others precisely undermining classical determinism; this would come across as an interesting paradox were it not for the fact that the outcome of these results was simply a redistribution of the existing models’ underbelly aimed at defining “chaos”.
In order to explain such paradox in this and the next paragraphs, we must first mention the fact that the Solar System, in all its complexity, shows a somehow
“regular” pattern due to the weakness of gravity and the total predictability of Kepler’s two-body problem. In view of this, Euler, Lagrange and Laplace studied increases in the amount of bodies in terms of changes in global stability due to small perturbations of a two-body problem, i.e., saw movement as an addenda to geometry, whereas Hamilton and Jacobi again added geometry as a factor to movement by describing dynamics as phase spaces whose volume was to be kept constant by the flow.
This cumulative intermingling of geometrical and dynamical outlooks was useful for a number of reasons, most importantly the reduction of the dimension via purposefully chosen symmetries, and a serious attempt was already being made late in the nineteenth century at finding corrections to Kepler’s problem by a third mass. There was a pending obstacle, though: proving the convergence of the resulting perturbed series. This problem was first glanced upon in the 1880s by K. T. W. Weierstrass who, with the aid of G. Mittag-Leffler and under the auspices of King Oscar of Sweden, favored the announcement of a prize in Acta Mathematica (volume 7, 1885/86) for finding the solution as a uniformly convergent series. The difficulty of finding this global solution as a series, let alone as a convergent one, is inferred from the revised draft of H. Poincar´e’s attempt which, although thwarted, won the prize and is nowadays considered landmark in the theory of Dynamical Systems. The problem as stated in the terms of the prize was finally solved, except for special cases, by K. F. Sundman in [136] for the Three-body Problem (Theorem 2.4.2) and by Q. D. Wang for the general N-Body Problem (Theorem 2.4.3). See [35] for details on the subject’s evolution from Weierstrass and Poincar´e’s “brilliant failure” onward.
The main success of Poincar´e’s work was his prediction of divergence as due to the presence of the so-called “small divisors”, ever since seen as a marker for the impossibility of predictions on the long-term evolution of systems as com- plex as the N-Body Problem. Since only a rarity of systems are significantly less complex than the latter Problem or happen to satisfy any of a wide list of requirements for integrability (some of which we will explain further on), the
study of small divisors began to focus, during the second half of the past century, on a very specific endeavor: the possible persistence, for a given perturbation of an integrable system, of certain traits or symptoms of integrability. Simply put:
how much of the geometry underlying dynamics prevails when perturbing an in- tegrable system into chaos. It was A. N. Kolmogorov who finally found, in 1954 [63], an answer to this question: to wit, that a somehow quantifiable majority of the trajectories of such non-integrable systems are quasi-periodical and may be computed through convergent expressions. V. I. Arnol’d and J. Moser, in the sixties, established further rigorous proofs of this fact, thenceforth known as the K.A.M. (Kolmogorov-Arnol’d-Moser) Theorem: besides [63], see [12], [14], [13]
and [102].
A long way has been travelled, and is still unconcluded, in order to detect and define chaos as an addendum to an ideal geometric groundwork – in other words, to obtain an axiomatic reassessment of our ignorance, couched on deter- ministic terminology. Although ancestry as defined by Ph.D. pupilship is not always determinant, it is indeed significant in this case that a direct line of such ancestry may be established from Gauss to Weierstrass and from Weierstrass to Kolmogorov and Moser, as well as from Kolmogorov directly to Arnol’d. See [33].
As for present and future sceneries, the main theme in the current study of chaos is the attempt at transversality between disciplines. In particular, the study of chaos from the algebraic point of view is a new, relatively recent trend estab- lishing direct continuity with the preceding and nowadays centered on two stages with more than a trait in common: the line of study initiated by S. L. Ziglin ([162], [163], see also [15]) and the one begun by J.J. Morales-Ruiz and J.-P. Ramis: see [93] and [95]. Ziglin’s theory relies strictly on the monodromy generators of the variational equations around a given particular solution, whereas Morales’ and Ramis’ theory uses linear algebraic groups containing the aforementioned mon- odromies and is naturally immersed in the Galois theory of linear differential equations, which we assume the reader is already familiar with – otherwise, see Section 2.2.2 of this thesis or [93] and [144] for the minimum necessary concepts.
1.3 Original results
Understandably, none of what has been said in the Section 1.1 seems susceptible of conclusive statements at this point, and what is explained in Section 1.2 is highly unlikely to be unified into a single theory in the short term. What is presented in this thesis, instead, is a compendium of algebraic non–integrability proofs for a short array of problems arising from Celestial Mechanics, the original Three-Body Problem among them, as well as a new necessary condition, stronger than mere integrability, which is applied to generalize some of the aforesaid proofs and may in turn be used for a wider class of Hamiltonian systems.
This is done in Chapters 3 and 4, after summarizing in Chapter 2 what is understood as (meromorphic) integrability in the Hamiltonian setting where these problems belong. This summary may also be seen as an introduction to some of the topics explained in Section 1.2.
1.3.1 Homogeneous potentials and N -Body Problems
Having Section 1.2 in mind, the N-Body Problem’s history of parallel attempts both at looking for new first integrals for it and proving it analytically or mero- morphically non-integrable should not come up as a surprise. Even less surprising is the partial success of the latter, especially in recent times thanks to the two parallel lines of study introduced in the last paragraph of Section 1.2. Using a consequence of the new theory by Morales-Ruiz and Ramis as applied to the fac- torization of linear operators, D. Boucher and J.-A. Weil ([23], [21]) proved the meromorphic non-integrability of the Three-Body Problem. Since the obstruction to integrability arising from the Boucher-Weil approach was precisely the presence of logarithms in the resulting decomposition, this may be seen as an instance of what was said in the last paragraph of Section 1.1. On the other hand, using the Ziglin approach, A. V. Tsygvintsev ([139], [140], [141], [142], [143]) proved the meromorphic non-integrability of the Three-Body Problem and ultimately set- tled the non-existence of a single additional meromorphic first integral except for three special cases (see Remark 3.3.1). It is finally worth noting that Ziglin ([164, Sections 3.1 and 3.2]) managed to settle strong conditions on the integrability of the Three-Body Problem and the equal-mass N-Body Problem.
Chapter 3 reobtains in simpler ways, strengthens and generalizes the results mentioned in the previous paragraph using the aforementioned theory started in [95] as applied to Hamiltonians of a specific kind: to wit, those which are classical with an integer degree homogeneous potential. Although conjectures and open problems will still prevail (see Chapter 5), the proofs given here are significantly shorter thanks to a significant step forward made in [95, Theorem 3].
Furthermore, using this same Theorem, a new necessary condition is established in Section 2.3.2 on the existence of a single additional integral for any classical conservative system – a condition in turn allowing us to discard the existence of an additional integral for the Three-Body Problem with arbitrary dimension and positive masses (a generalization of Bruns’ Theorem 2.4.5, that is) and for the planar N-Body Problem with equal masses if N = 4, 5, 6. It must be said that, in the equal-mass case, the only apparent obstacle keeping us from extending Bruns’
to an arbitrary amount of bodies was a technical one, namely the structure of a certain algebraic extension of the Nth cyclotomic field for general N ≥ 7.
Specifically, the new results in Chapters 2 and 3 are Theorem 2.1.10 and Corollary 2.3.5, as well as Theorems 3.2.2, 3.2.3 and 3.3.10 and Corollary 3.3.11, as well as Lemmae 3.3.5 and 3.3.6. The Lemmae used in their proofs are mostly a reformulation of known previous results and would hardly qualify as new, al- though special mention may be made of Lemmae 2.1.7 and 2.1.8. All of the open problems in Chapter 5 find numerical evidence in their favor, gathered for a widespread family of values of N. This is true both for the equal-mass Problem and for a fairly large variety of masses. A word may be said about the impending publication of part of these new results in [99].
1.3.2 Non-integrability of Hill’s Problem
Hill’s Lunar Problem appears in Celestial Mechanics as a limit case of the Restricted Three-Body Problem, itself a special instance of the problem in the previous paragraph for N = 3. Moreover, and aside from the fact that it appears to be the simplest illustration of gravitational dynamics with more than two bodies, Hill’s problem provides with information in turn casting light on several other problems in Celestial Mechanics. It contains no parameters and is globally far from any simple well–known problem. Strong numerical evidence of its lack of integrability has been given in the past, although no rigorous proof in this respect had been done in general terms up to this thesis.
In Chapter 4, an algebraic proof of meromorphic non–integrability is presented for Hill’s Problem which, rather than exploiting the tools used and found in Chapter 3, avails itself of the deep-set theoretical basis of those tools – not only out of willful diversification, but also because those previous tools were not enough for our purpose. Beyond the novelty of the result itself, thus, Chapter 4 stands as an example of the adequacy of the most general instance of Morales’ and Ramis’
theory to many significant problems – an instance with whose aid we identified the concrete contributions, embodied in special functions, which probably made this proof so hard to find in the past. Hence, in all its surgical detection of obstructions to integrability, this is one of the places where the thesis is closest to echoing the second paragraph in Section 1.1 without fully conveying it.
All of the Lemmae and Theorems in Chapter 4, that is, those stated in Sub- section 4.1.1 (Lemmae 4.1.1 and 4.1.2 and Theorem 4.1.3, and the immediate consequence given in Corollary 4.1.4) are new results. As opposed to the previ- ous Chapter, all that is said in Chapter 4 has already been published, in [98], in a joint work with the advisors of this present thesis.
1.4 General structure, notation and conventions
This thesis consists of four chapters. There will be only one figure derived from numerical simulation (see Section 4.4), since we intended to lean as little as possible on numerical results and only used them for illustration purposes. The first and last chapters will be mainly a compendium of known information except for a new result in Section 2.3.2 and an ensemble of conjectures in Chapter 5.
There will be a subject index at the end, in which page numbers will be marked in boldface if the word is defined in the given page, and in regular face if said word is simply mentioned.
Given a field K and a K-vector space V of finite dimension n, EndK(V ) will denote the space of endomorphisms f : V → V (as opposed to other notations such as LK(V ; V ) or HomK(K; K)) and, given n ∈ N, Mn(K) will be the alternative way of writing the ring EndK(Kn) of all square n × n matrices with their entries in K. Similarly, GL (V ) ⊂ EndK(V ) will be the group of invertible linear transformations and, fixing bases in Kn, the group will be immediately identified with that of invertible n×n matrices and written GLn(K); the subgroup of GLn(K) comprised of linear transformations whose determinant is equal to the unit element of the group (K∗, ∗) is denoted as SLn(K). On(R) will in turn stand
for the set of orthogonal matrices with their entries in R, and Sp2n(K) will stand for the symplectic group of degree 2n over K. Although the underlying set will be a cartesian product in both cases, direct sums will be written differently for algebraic groups G1, . . . , Gn (see Section 2.1) and K-vector spaces V1, . . . , Vn: G1× · · · × Gn and V1⊕ · · · ⊕ Vn, respectively.
There will be a number of cases in which the above field K will be C by de- fault. This will be the case for vector and matrix functions, for instance, unless stated otherwise. All vectors will be denoted in boldface and their norms will be written in ordinary face. All norms will be assumed Euclidean by default, for it is through these that the N-Body Problem finds its simplest known formulation.
For every vector whose entries are likely to be broken down in separate vectors of lesser size, at most two different boldface types will be used, albeit with the same letter: for any n, m ∈ N, a vector in Cnm will be written with italic boldface, q (its norm being q) if the n consecutive m-vectors making up for its entries are also being considered; in such case, these latter will be written in regular boldface, q1, . . . , qn ∈ Cm, their norms written as q1, . . . , qn, respectively. If further hierar- chy is needed, we will maintain either italic or regular boldface. Vectors will be freely written in concatenation, e.g. zT = qT, pT
= qT1, . . . , qTn, pT1, . . . , pTnT
, but we will avoid the T superindex unless we have to make specific reference to scalar products, e.g. in Rayleigh quotients. Boldface as described in all of the above considerations will be applied exclusively to constant vectors and vector functions of one variable, e.g. q = q (t), whereas vector functions with more than one argument, e.g. f = f (t, q), will be written in regular face.
Matrices will be written in capital letters, whether Latin or Greek. Be it for matrices or for vectors, notation will be sometimes implicit by means of subindexes, e.g. (bi,j)i,j=1,...,n may stand for B ∈ Mn(K) and (ai)i=1,...,n may stand for a vector a ∈ Kn; the terms inside the parentheses will occasion- ally stand for whole vectors or matrix blocks instead of single entries. Square roots for diagonal matrices will be defined as usual whenever the original di- agonal entries are real and non-negative: M1/2 = diag√mi,i : i = 1, . . . , n
if M = diag {mi,i : i = 1, . . . , n} ∈ Mn. As for vector functions of one variable, x : X ⊂ K → Kn, we will occasionally write them as Cartesian products, e.g.
x = x1 × · · · × xn, whenever further reference to their coordinate functions is pertinent.
Since there will only be one independent variable t properly regarded as time, an overdot will stand for dtd all through the text and(k) will stand for dtdkk, k ≥ 4, whereas ′ will usually imply derivation with respect to phase variables of Hamil- tonian systems. It is worth noting this time variable t will be complex by default all through the text. Γ will often stand for Riemann surfaces, and P1 will always stand for the (complex) projective line.
Defining the Kronecker delta δi,j as usual, n
en,k = (δi,k)Ti=1,...,no
will be the canonical basis for Rn. Zero vectors and zero and identity matrices will be written with their dimension as a subindex whenever deemed necessary, e.g. 0n ∈ Kn or 0n×n, Idn∈ Mn(K). |·| will denote absolute value or modulus indistinctively.
√−1 = i will always be denoted in Roman, non-italic font. The consideration of points in the plane as either complex numbers or real 2-vectors will also be
tacit depending on the context. The determination for complex square roots will be that given by the analytic continuation of the positive real square root, i.e.
√z :=√rei2θ whenever z = reiθ and θ ∈ [0, 2π].
Theoretical background
This chapter is devoted to a concise introduction to the theoretical tools used for our main results. Despite its mainly expository nature it contains a new result, proven in Subsection 2.3.2. Basic knowledge will be assumed from the reader concerning complex functions, differential systems, calculus on manifolds, differ- ential forms, group actions, representation theory and invariant theory; readers not acquainted with these themes may first read [2], [3], [13], [55], [68], [93], [131], and [145]. All through the rest of the text, we shall make no significant forays into the topics of special functions, representation theory, Algebraic Geometry and Celestial Mechanics other than the ones made in this chapter.
2.1 Useful results from Algebraic Geometry
See [19], [55], [68], [93], [127] or [131] for technical details and further information.
2.1.1 Preliminaries
From now on, each group G will have its unit element written as eG, subindex G
being dropped for the most part. We recall calling a subgroup H ⊂ G normal if, for every x ∈ G, xHx−1 = H. It is straightforward to establish that the kernel of any group homomorphism, as well as the image of a normal subgroup under an epimorphism is always a normal subgroup of the source group. A sequence of subgroups
G = G0 ⊃ G1 ⊃ · · · ⊃ Gm, (2.1) for any given m ∈ N, is called a tower of subgroups. Tower (2.1) is called normal if Gi+1 is a normal subgroup of Gi for each i = 0, . . . , m − 1. A group G is called solvable if there is at least one m ∈ N such that G has a normal tower (2.1) in which Gm = {eG}. It is a known fact that given a normal subgroup H ⊂ G then G is solvable if and only if H and G/H are solvable; in particular, f : H → H′ = f (H) given, ker f is a solvable normal subgroup and thus H/ ker f ≃ H′ is solvable as well, meaning: solvability is preserved under group epimorphisms.
Given a finite-dimensional vector space V over an algebraically closed field K, let S be a finitely-generated K-algebra of K-valued functions on V . Two such algebras are:
9
1. the K-algebra K [V ] of polynomial functions on V , i.e. functions of the form f = P ◦ ϕ : V → K, P : Kn → K being a polynomial, P ∈ K [x1, . . . , xn], and ϕ being an isomorphism between V and Kn;
2. and the quotient field of K [V ], i.e. the K-algebra K (V ) of rational func- tions defined on V , i.e. functions of the form f = F ◦ ϕ : V → K, F : Kn→ K being a quotient of polynomials, P (x1, . . . , xn) /Q (x1, . . . , xn) with P, Q ∈ K [x1, . . . , xn], and again ϕ being an isomorphism between V and Kn.
If S = K [V ] it may be easily proven (e.g. [68, Proposition 5.2 (Chapter 10)]) that the sets Z (I) of zeros of ideals I ∈ S are affine varieties over K ([55, §1.1]) and thus closed sets of a certain topology called the Zariski topology ([55, §1.2]).
For the remainder of this Section, any reference to topology will be henceforth set exclusively in either the Zariski topology or the one therefrom induced on subsets or cartesian products.
We recall a topological space X is irreducible if two non-empty open subsets of X have a non-empty intersection. In the next results, as said in the previous paragraph, subsets X ⊂ V will be systematically endowed with the subspace topology induced by the Zariski topology of V . It is easy to establish that V is irreducible ([131, Corollary 1.3.8]) and thus:
Lemma 2.1.1. Any non-empty open set A ⊂ V is dense in V .
2.1.2 Linear algebraic groups and Lie algebras
Linear algebraic groups
Recall an algebraic group over K as being an affine algebraic variety over K endowed with a group structure such, that the two maps µ : G × G → G, ι : G → G defined by µ (x, y) = xy and ι (x) = x−1 are morphisms of varieties.
In particular, a special type of algebraic group is a linear algebraic group which is defined as a Zariski closed subgroup of some GL (V ), V being finite- dimensional K-vector space as above. We also recall ([55, §7.4]) a morphism of algebraic groups as being a group homomorphism φ : G → G′ which is also a morphism of varieties; whenever G′ = GLn(K) we say morphism φ is a (rational) representation; in light of this, it is usually advisable to view GL (V ) as an algebraic group all its own, specifying its Zariski topology in an unambiguous way by any arbitrary choice of basis for V ≃ Kn since any such choice in Kn corresponds to an inner automorphism x 7→ yxy−1 in GLn(K).
Since the product topology in G1× · · · × Gn is precisely the initial topology with respect to projection maps πi : G → Gi defined by πi(g1, . . . , gn) := gi, each of these projections will be continuous with respect to the Zariski topology in G. In particular, if G1, . . . , Gn are algebraic groups, then for any connected subgroup H ⊂ G1× · · · × Gn each image πi(H), i = 1, . . . , n, is a connected subgroup of Gi with respect to the Zariski topology in Gi.
A representation is called faithful if it is injective. Given any representation φ : G → GL (V ) of an algebraic group G, the operation
G × V, (x, v) 7→ x · v := φ (x) v,
is clearly a group action of G on V . In this case V is usually called a (rational) G-module. For any algebraic group G acting over V , we call Gv = O (v) = {g · v : g ∈ G} the G-orbit of v ∈ E. G-module V is called faithful if (x, v) 7→
x · v is faithful as a group action, i.e. if φ is a faithful representation. Module V is called irreducible if it has exactly two submodules: {0} and V itself. More generally, a finite-dimensional G-module V is completely reducible if for every submodule V1 ⊂ V there is another submodule V2 ⊂ V such that V = V1⊕ V2 or, equivalently, if V is the direct sum of some of its irreducible submodules.
Given an algebraic group G, the identity component G0 of G is the unique (topologically) irreducible component containing eG. Any algebraic group has a unique largest normal solvable subgroup, which is automatically closed ([55, Corollary 7.4 and Lemma 17.3(c)]). Its identity component is thus the largest connected normal solvable subgroup of G; it is called the radical of G and denoted R (G). The subgroup of R (G) consisting of all its unipotent elements (i.e., those elements expressible as the sum of the identity and a nilpotent element) is normal in G; it is called the unipotent radical ([55, §19.5]) of G, denoted as Ru(G), and may be characterized as the largest closed, connected, normal subgroup formed by unipotent elements of G. If R (G) is trivial and G 6= {e} is connected, G is called semisimple; this is the case, for instance, for SLn(K) ([55, §19.5]). If G is semisimple, then every G-module V is completely reducible. G is furthermore called simple if it has no closed connected normal subgroups other than itself and {e}; SLn(K) is again a valid example ([55, §27.5]).
Lie algebras
Everything defined and asserted in this Subsection is found and verified in detail in [19, Chapter 1, from §3 onward], [55, Chapters 9 and 10], [93, Chapters 2, 3 and 4] or [106, Chapters 1 and 3].
A Lie algebra over K is a particular kind of algebra over a field; it is defined as a K-vector space a together with a bilinear binary operation [·, ·] : a × a → a, called the Lie bracket, such that [x, x] = 0 for all x ∈ a and the Jacobi identity holds:
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] , x, y, z ∈ a.
Lie subalgebras will be accordingly defined as subspaces of a Lie algebra which are closed under the Lie bracket. An ideal of the Lie algebra a is a subspace h of a such that [a, x] ∈ h for all a ∈ g and x ∈ h. All ideals are trivially subalgebras, although the converse is not always true.
The commutator series of a Lie algebra a, sometimes also called the de- rived series, is the sequence of subalgebras recursively defined by ak+1 :=
ak, ak
, k ≥ 0, with a0 := a. A Lie algebra a is solvable if its Lie algebra commutator series
ak
k vanishes for some k. a is simple if it is not abelian and has no nonzero proper ideals; it is straightforward to prove that solvable implies not simple for any Lie algebra. A Lie algebra is semisimple if it is a direct sum of simple Lie algebras.
Let G be an algebraic group over C; since, being an affine variety, it may be endowed with the usual complex topology as well as with the Zariski topology,
it is actually a Lie group ([106, §1 (Chapter 1)]), i.e. a group which is also a differential manifold, such that the group operations are compatible with the differential structure. To every Lie group G we can associate a Lie algebra (whose indication in black letters, g, is usually the only change in notation), in a way completely summarizing the local structure of the group; the underlying vector space of g is the tangent space of G at the eG, and we can heuristically characterize all elements of the Lie algebra as elements of G which are “infinitesimally close”
to eG. We will usually call g the Lie algebra of G, writing it alternatively as Lie (G). See [55, Chapter 1] for concise definitions and properties. It is also reasonably immediate to prove that the Lie algebra of a semisimple algebraic group is semisimple itself.
We have the following result (see also [93, Proposition 2.2]):
Lemma 2.1.2. sl2(C), i.e. the Lie algebra of SL2(C), has no simple subalgebras other than itself.
Proof. Indeed, the dimension of sl2(C) is three, and thus any proper subalgebra of sl2(C) should be of dimension smaller than or equal to two; all such subalgebras are solvable ([93, §2.1]), thus not simple.
2.1.3 Rational invariants
Let G ⊂ GL (V ) be a linear algebraic group. We may define, as is done in [93,
§4.2], the action of G on C [V ] or C (V ):
g · f := f ◦ g−1, g ∈ G, f ∈ C (V ) .
We define by C [V ]G(resp. C (V )G) the C-algebra of G-invariant elements of C [V ] (resp. C (V )); hence the denomination rational invariant for any f ∈ C (V )G. We may furthermore assume G is connected, since G has an invariant if, and only if, G0 has an invariant; this fact, which is a consequence of the finite index of G0 in G, may be found proven in the first Lemma of [165, Chapter 1]; see also [15].
For any subgroup G of GL (V ), e.g. a linear algebraic group acting over V , the set of G-orbits of G is clearly a partition in V . Moreover, given an algebra of C-valued functions S and a function α which is invariant by G, e.g. S = C (V ) and α ∈ C (V )G, the restriction of α to each of the orbits of G is constant.
Furthermore, if G has a non-empty open orbit O, then any invariant of G is constant on O and by extension and the density of the latter (due to Lemma 2.1.1) renders α constant on the whole space V . Thus, algebraic groups with an open non-empty orbit do not have non-trivial rational invariants, i.e., their only rational invariants are constants. Conversely, we have the following:
Lemma 2.1.3. Let G be an algebraic subgroup of SL2(C) with no non-trivial rational invariants with the natural representation of G on C2. Then, G has an open orbit.
Proof. As is well-known, the only algebraic subgroups of SL2(C) with no non- trivial rational invariants are
H :=
λ 0 µ λ−1
, λ ∈ C∗, µ ∈ C
,
and SL2(C) itself. H has the open orbit (C \ {0}) × C and SL2(C) has the open orbit C2\ {0}.
In the following three results, m will be assumed to be an arbitrary natural number.
Proposition 2.1.4. Let G = G1× G2× · · · × Gm, Gi being an algebraic subgroup of SL2(C) for each i = 1, ..., m. If G has a (non-trivial) rational invariant for the natural representation of G on (C2)m, then Gi must have a non-trivial rational invariant for at least one i.
Proof. Assume each Gi has no non-trivial invariants; then, it has an open orbit Oi. Thus G has an open orbit O1× · · · × Om and reductio ad absurdum yields the result.
Corollary 2.1.5. Let G = G1 × G2× · · · × Gm, Gi being an algebraic subgroup of SL2(C) for each i = 1, ..., m. If G has a non-trivial rational invariant, then Gi has a commutative identity component G0i for at least one i.
Proof. In virtue of the classification of the linear algebraic subgroups of SL2(C) ([93, Proposition 2.2]) we know that an algebraic subgroup H of SL2(C) has non- trivial rational invariants if and only if the identity component H0 is commutative and the result follows from Proposition 2.1.4.
Corollary 2.1.6. SL2(C)m has no non-trivial rational invariants. Lemma 2.1.7. Let g be a simple Lie subalgebra of Ln
i=1sl2(C) = Lie (SL2(C)n).
Then g ≃ sl2(C).
Proof. For each i = 1, . . . , n let
πi|g: g → sl2(C) , (x1, ..., xn) 7→ xi, be the restriction of the canonical projection πi : Ln
i=1sl2(C) → sl2(C) to g. There is at least one i such that πi|g(g) 6= {0}, since each element x = (x1, ..., xn) ∈ g is precisely equal to (π1(x) , ..., πn(x)), and were πi|g ≡ {0}, i = 1, ..., n, we would then have g = {0}. Thus, there is at least one i for which πi|g has a non-trivial image πi(g) 6= {0}, itself a subalgebra of the Lie algebra sl2(C) which admits no simple subalgebras other than itself, as said in Lemma 2.1.2; this latter fact implies πi(g) = sl2(C) ≃ g/ ker πi|g. But g is simple as well, and thus the ideal ker πi|gmust be either {0} or g. It is clear that ker πi|g= {0}, since ker πi|g = g would imply sl2(C) ≃ g/ ker πi|g = {0} which is obviously absurd.
Lemma 2.1.8. Let G be an algebraic group and V a G-module such that G is faithfully represented as a subgroup of SL2(C)n,
ρ : G → SL2(C)n. Assume πi(G) = SL2(C) for i = 1, . . . , n,
πi : SL2(C)n → SL2(C) , (A1, . . . , An) 7→ Ai,
being the i-th projection for each i = 1, . . . , n. Then, the Lie algebra g of G satisfies g ≃Lm
i=1sl2(C) for some m ≤ n.
Proof. The hypotheses imply V is a completely reducible G-module. In order to further prove G semisimple, let us assume the contrary, i.e. that R (G) 6= {e};
then not every πi(R (G)) would be nontrivial since ρ is injective and thus so is ρ|R(G), i.e. R (G) is represented faithfully as a subgroup of SL2(C)n: R (G) ֒→
π1(R (G)) × · · · × πn(R (G)) ⊂ SL2(C)n. But this is absurd since πi(R (G)) is trivial, i = 1, . . . , n; indeed, each πi(R (G)) ⊂ SL2(C) is a normal, connected, solvable subgroup of a simple algebraic group since πi is a group epimorphism and SL2(C) is simple. Thus, πi(R (G)) = {Id2} for each i = 1, . . . , n implying R (G) = {e}, i.e. G is a semisimple algebraic group. Let g := Lie (G) the corresponding semisimple Lie algebra and g = g1⊕ · · · ⊕ gm a decomposition in simple algebras. From Lemma 2.1.7, we know
gi ≃ sl2(C) , i = 1, . . . , m, and thus g ≃Lm
i=1sl2(C).
If G is a semisimple algebraic subgroup of SL2(C)n, it is in particular a subset of the symplectic group of a symplectic C-vector space E ≃ C2n, since SL2(C)n ⊂ Spn(C); Lemma 2.1.8 assures g = Lie (G) ≃ Lm
i=1sl2(C) for some m ≤ n and, in virtue of this, we have
g≃ Mm
i=1
sl2(C) ⊂ Mn
i=1
sl2(C) ⊂ spn(C) ≃ S2E∗, {·, ·} ,
the latter isomorphism of Lie algebras being proven in [93, Lemma 3,2], E∗ being the dual C-space of E, SkE∗ being the symmetric algebra on E∗ (that is, the ring of homogeneous quadratic Hamiltonian functions defined over E giving rise to linear, constant-coefficient Hamiltonian fields) and {·, ·} being the Poisson bracket introduced, for instance, in Section 2.2.1 below; see [93, §3.1, 3.4] for more details.
We say that a subalgebra g ⊂ spn(C) ≃ (S2E,∗{·, ·}) has a rational invari- ant α ∈ C (E) if {g, α} ≡ 0. The following is straightforward to verify; see for instance [93, §4.2]:
Lemma 2.1.9. An algebraic group G has a non-trivial rational invariant if, and only if, Lie (G) has a non-trivial rational invariant.
So far we have proven the following train of implications:
1. (Lemma 2.1.6) SL2(C)m has no non-trivial rational invariants;
2. therefore, in virtue of Lemma 2.1.9, Lie (SL2(C)m) = Lm
i=1sl2(C) has no non-trivial rational invariants;
3. thus, for any linear algebraic group G satisfying the hypotheses of Lemma 2.1.8, and in virtue of the latter, Lie (G) = g ≃ Lm
i=1sl2(C) has no non- trivial rational invariants;
4. hence, again in virtue of Lemma 2.1.9, G has no non-trivial rational invari- ants.
In other words: we have just proven the following:
Theorem 2.1.10. Let G ⊂ SL2(C)n be an algebraic group such that the pro- jections πi(G) = SL2(C), i = 1, ..., n. Then, G has no non-trivial rational invariants.
Theorem 2.1.10 will be of key importance for the new result (Corollary 2.3.5) proven in Section 2.3.2.
2.2 Notions of integrability
As said in Section 1.1, specialization is the most immediate symptom in the study of integrability of any given system
˙
y= f (t, y) , y= y1× y2× · · · × yn : C → Cn. (2.2) The two distinct notions described in this section, adapted to two precise types of dynamical systems, do have a common trait, though: the ability to perform integration by quadratures, that is, to express the general solution as an
“elementary” function of a finite nested sequence of integrals of “elementary”
functions, constants of integration being the parameters of the solution manifold.
See [113] for a wider outlook on the subject.
2.2.1 Integrability of Hamiltonian systems
Let us restrict our attention to a very special example of such a system as (2.2).
Everything explained here can be found in more detail in [13], [15], [75], [89], [132], [145], and especially [18], [65] and [93].
All assertions and definitions in this Section, save for the hypotheses of The- orem 2.2.2, are made in the complex setting as done in [93, Chapter 3] and throughout [95]. Similar assertions and definitions adjusted to real bundles and fields may be found in [14], [13], [18] and especially [75].
A symplectic manifold is a complex manifold of even dimension 2n along with a nondegenerate closed 2-form Ω, called the symplectic form, whose non- degeneracy allows the definition of a musical isomorphism of vector bundles,
♭ : T M → T∗M, ♭X = Ω(X).
These manifolds arise naturally as phase spaces of the class of differential systems we are now introducing.
A Hamiltonian vector field is a field XH defined on the symplectic manifold M, such that XH = ♭−1· dH for some function H, usually called the Hamilto- nian. The differential equation satisfied by the integral curves of a Hamiltonian vector field is called a Hamiltonian system; in virtue of Darboux’s theorem ([18, Theorem 1.1], [93, Theorem 3.1]), it may be written, in canonical local coor- dinates (q, p) = (q1, . . . , qn, p1, . . . , pn) (referred to as positions and momenta, respectively), in the following form
˙qi = ∂H
∂pi
, ˙pi = −∂H
∂qi
, i = 1, . . . , n; (2.3)
one usually calls these (which will stand for (2.2)) Hamilton’s equations as- sociated to Hamiltonian H. They may also be written as ˙z = XH(z), noting z := (q1, . . . , qn, p1, . . . , pn). This context also allows the definition of canonical transformations, i.e. changes of the variables z under which the symplectic form remains invariant; in other words, under which the Hamiltonian form of the equations is maintained for arbitrary Hamiltonians. Furthermore, the musical isomorphism ♭ allows the adjunction of a Poisson algebra structure on M, bear- ing Poisson brackets {f, g} = Ω (Xf, Xg) which in canonical coordinates may be expressed as {f, g} =Pn
i=1
∂f
∂pi
∂g
∂qi −∂q∂fi∂p∂gi
. The following holds:
Proposition 2.2.1. f is a first integral (that is, a function constant over inte- gral curves) of XH if, and only if, {H, f} = 0 (i.e. H and f are in involution, or commute). In particular, H is always a first integral of XH.
Whenever the idiom additional first integral appears, it will be referring to one which is independent and in involution with a certain known set of m < n first integrals, be it a singleton F = {H} as is the case of Hill’s Problem (see Section 2.4.2 and Chapter 4), or the set F of 12(d + 2) (d + 1) “classical” integrals for the d-dimensional N-Body Problem (see Section 2.4.1 and Chapter 3).
The following result does not merely provide some Hamiltonians a description of their phase spaces; in most cases, it also confers the whole area a precise notion of integrability; for further details and a proof, see [13, Chapter 10: §49 and §50]
or [18, Theorem 1.2 and the remainder of §1.4]. Let XH be an n-degree-of-freedom real Hamiltonian.
Theorem 2.2.2 (Liouville-Arnol’d). Assume XH has n functionally independent first integrals f1 = H, f2, . . . , fn in pairwise involution. Let a ∈ Rn and
M (a) = {z : fi(z) = ai, i = 1, . . . , n}
be a non-critical level manifold of f1, . . . , fn. Then, 1. M (a) is an invariant manifold of XH;
2. if compact and connected, M (a) is diffeomorphic to Tn= Rn/Zn, and in a neighborhood of the former there exists a coordinate system (I, φ) ∈ Rn×Tn in which (2.3) read
I˙i = 0, φ˙i = ωi, i = 1, . . . , n,
with ωi = ωi(I) , i = 1, . . . , n. In particular, XH can be integrated by quadratures.
Directly after the sufficient condition provided by Theorem 2.2.2,
Definition 2.2.3. We call system (2.3) integrable in the sense of Liouville- Arnol’d, completely integrable or simply integrable, and extend this def- inition to XH and H, if (2.3) has n functionally independent integrals f1 = H, f2, . . . , fnin pairwise involution. {f1, . . . , fn} is usually called a complete set of independent first integrals.
We can generalize this definition by allowing a lower cardinality for the set of additional integrals:
Definition 2.2.4. We call the Hamiltonian partially integrable if there is a set of 0 < l < n additional first integrals in pairwise involution.
Obviously, denying Definition 2.2.4 for a given value 0 < l < n (as will be the case in part of Chapter 3 for l = 1) implies denying Definition 2.2.3. This fact also plays a pivotal role in Subsection 2.3.2 below.
Given a Hamiltonian XH, there is a number of ways of searching for additional first integrals, although none of them works for all cases – see [51] for more details.
One of these ways is using Theorem 2.2.2 directly, i.e. looking for solutions f to the partial differential equation {H, f} = 0. For exceptional examples in which this method works, see for instance [100] and [101], both owing to the basic work [116] about generalized Noether symmetries. One may also pursue the so-called integrability in the sense of Hamilton-Jacobi, i.e. the possibility of finding some explicit canonical variables sj, rj, j = 1, . . . , n separating the Hamilton-Jacobi equation for the action S (see [13, Chapter 10] or [112, Chapter 3]). See also [5]
for extensive information on algebraic integrability, in turn related to embeddings of abelian varieties in an affine space through a reedition of ideas by Kowalevskaia and Painlev´e.
Remarks 2.2.5.
1. It is explicitly assumed that the first integrals sought after, both in Theorem 2.2.2 and in Definitions 2.2.3 and 2.2.4, are defined globally; that is, in no way are we referring to the local integrals existing trivially in virtue of Cauchy’s Theorem.
2. Although we restricted everything to R, Hamiltonian formulation may also be defined in the complex setting by allowing t and z to be complex-valued and functions and vector fields to be analytical or meromorphic. The only nuisance to some purposes, though, is the absence of a complex analogue to Theorem 2.2.2 except for special cases (see [93]). The usual procedure is to work with complex meromorphic Hamiltonians which restrict to real for real dependent and independent variables, observing Definitions 2.2.3 and 2.2.4 on the real system and then complexifying all variables.
3. From now on, and in tune with what has been said in item 2, whenever we refer to Hamiltonian integrability we will refer to meromorphic integra- bility: additional first integrals, whether in Theorem 2.2.3 or 2.2.4, will be assumed to be meromorphic along a subset of a complex manifold. Given a domain Ω in Cn or any n-dimensional complex manifold, and complex- analytic subset of dimension n − 1 (or empty) P ⊂ Ω, we recall a function f defined on Ω\P is meromorphic if for every p ∈ P there is a neighborhood U ⊂ Ω of p and functions φ, ψ holomorphic on U without common non- invertible factors in the ring O (U) of holomorphic functions on U, such that f ≡ φ/ψ on U \ P . See also [48, Chapter 8, p. 246] for a precise definition in the context of sheaf theory.
2.2.2 Integrability of linear differential systems
The concept of integrability for linear homogeneous differential equations is con- ventionally limited to the possibility of finding their general solution in terms of algebraic functions, integrals and exponentials of known functions or any finite combination of all three. This second notion is naturally inscribed in differential Galois theory as will be seen in Definition 2.2.14 and Theorem 2.2.15. Every single fact stated here is described in detail in references [93, Chapter 2], [95, Section 3] and [144, Chapter 1] and, to a lesser degree, Sections 2 and 3 in [15];
Chapters 1 through 6 in [78] may also be useful.
Definition 2.2.6. Let K be a field. A derivation on K is an additive map
∂ : K → K satisfying the Leibnitz rule ∂ (ab) = ∂ (a) b + a∂ (b) , a, b ∈ K. A differential field is a pair (K, ∂K) consisting of a field and a derivation on it.
Definition 2.2.7. An extension of differential fields, usually noted L | K, is an inclusion L ⊃ K such that ∂L|K ≡ ∂K.
(K, ∂K) given, we henceforth note ∂ = ∂K unless necessary, and use this notation for elements of Kn extending the derivation entrywise. However, we will avoid the notation so frequent in most texts on Galois differential theory a′ = ∂ (a) so as to be consistent with what was said in Section 1.4.
Definition 2.2.8. The constants of a differential field (K, ∂) are the elements of the subfield Const (K) := ker ∂ of K.
All fields and extensions will be assumed to be differential from this point on. We assume characteristic zero for every field considered. The set of all K- automorphisms of any differential extension L | K, (i.e., field isomorphisms σ : L → L such that σ|K ≡ IdK and ∂ ◦ σ ≡ σ ◦ ∂) is a group under map composition and will be denoted by AutK(L). Given any m ∈ N, and using the propagation of morphism axioms of any σ ∈ AutK(L) to elements of Mm(K),
(σai,j)1≤i,j≤m(σbi,j)1≤i,j≤m = Xm
r=1
σ(ai,r)σ(br,j)
!
1≤i,j≤m
= σ
Xm r=1
ai,rbr,j
!
1≤i,j≤m
,
we will indulge in as many abuses of notation as necessary when extending σ entrywise to any m × m matrix.
Given a linear homogeneous differential system
∂y = Ay, A ∈ Mn(K) , (2.4)
and an extension E | K containing a set V of solutions of (2.4), there is always a minimal differential subfield L ⊂ E containing both K and the entries of the elements of V ; we write L = K (V ) and say L is generated over K as a dif- ferential field by the entries of elements of V and using (2.4). Since (2.4) is linear and homogeneous, V is a Const(L)-vector space of dimension at most n. AutK(L) preserves V and acts on it as a group of linear transformations over Const(K), and if Const(L) = Const(K) the restriction of AutK(L) to V gives a
faithful representation AutK(L) → GL(V ). V owes its relevance to those situa- tions in which it is precisely defined as the maximal set of linearly independent solutions of (2.4), thus establishing the differential analogue of a Galois exten- sion; such an analogue corresponds to the case dimConst(L)(V ) = n and actually matches the situation in which no new constants are added to K:
Definition 2.2.9. L | K is a Picard-Vessiot ( P–V) extension for (2.4) if 1. Const(L) = Const(K);
2. there exists a fundamental matrix Φ ∈ GLn(L) for the equation; and 3. L is generated over K as a differential field by the entries of Φ and using
(2.4).
Given a P–V extension L | K for (2.4) and an intermediate extension L ⊃ L1 ⊃ K then L | L1 is also a P–V extension for some linear ordinary differential system over L1. We are calling L | K a Picard-Vessiot extension if it is P–V for some linear ordinary differential system over K; an intrinsic definition may indeed be made, regardless of the equation. For the sake of simplicity and concre- tion we are henceforth assuming all fields considered have C as field of constants.
This assumption also assures existence and uniqueness of P–V extensions.
An essential property of P–V extensions is normality:
Lemma 2.2.10. For any a ∈ L \ K, there is a differential K-automorphism σ of L such that σ (a) 6= a.
Definition 2.2.11. If L | K is a P-V extension for (2.4), then AutK(L) will be denoted Gal (L | K) and called the Galois differential group of L | K (or of (2.4)).
The Galois differential group of an equation (2.4) is a linear algebraic group;
indeed, given a fundamental matrix Φ ∈ GLn(L), σ (Φ) is also a fundamen- tal matrix and hence σ(Φ) = ΦR(σ) with R(σ) ∈ GLn(C), which yields an n-dimensional faithful representation
ρ : Gal (L | K) → GLn(C) , σ 7→ R (σ) ; (2.5) this renders Gal (L | K) a linear group. For a proof of its being also Zariski closed, i.e. a linear algebraic group, see [144, Theorem 1.27]. Furthermore, the monodromy group of an equation (2.4), attained through analytical continuation of solutions, is a (generally not Zariski closed) subgroup of the differential Galois group of the corresponding P–V extension. Whenever G is the differential Galois group of some P–V extension, we are identifying elements σ of G with the cor- responding matrices R(σ) defining representation ρ in (2.5). In other words, we will be dealing indistinctively with the linear algebraic group G and the matrix group ρ(G).
Remark 2.2.12. Let G be the Galois differential group of the juxtaposition of uncoupled linear differential systems,
∂y = diag (A1, A2, . . . , Am) y, Ai ∈ Mni(K) , i = 1, . . . , m (2.6)
each subsystem ∂yi = Aiyi having Galois differential group Gi for i = 1, . . . , m.
Then, G is a linear algebraic subgroup of the direct product G1×· · ·×Gm, as may be easily established from the propagation of morphism axioms to matrix blocks (hinted at right after the above definition of K-automorphisms) and the fact that block-diagonal differential system (2.6) admits identically block-diagonal funda- mental matrices Φ = diag (Φ1, . . . , Φm) and thus just as identically block-diagonal matrix representations (2.5) of the elements of G; it is also straightforward to further prove that if π1, . . . , πm are the usual projections of G1× · · · × Gm, then πi(G) ≃ Gi for each i = 1, . . . , m; see [93, Chapter 2] for details as written in a synthetic, coordinate-free formulation.
We now state the so-called Fundamental Theorem of differential Galois theory.
Theorem 2.2.13. Let L | K be a Picard-Vessiot extension with common field of constants C, and let G = Gal (L | K), S the set of closed subgroups of G and L the set of differential subfields of L. Define
α : S → L, α (H) = LH,
(LH being the subfield of L formed by H-invariant elements); and β : L → S, β (L1) = Gal (L | L1) ,
Gal (L | L1) being the subgroup of G of L1-linear differential automorphisms of L. Then,
1. α i β are mutual inverses;
2. the following are equivalent,
(a) H ∈ S is a normal subgroup of G;
(b) L1 := α (H) = LH is a P–V extension of K;
and in such case Gal (L | L1) = H and Gal (L1 | K) ≃ G/H.
As foretold at the start of this subsection, we are now introducing the strict definition of what is to be called an integrable linear differential equation; it is precisely one whose P–V extension falls into the following category:
Definition 2.2.14. Let K be a differential field. L | K is called a Liouville extension if no new constants are added and there exists a tower of extensions
K = L0 ⊂ L1 ⊂ · · · ⊂ Ln = L (2.7) such that for i = 1, . . . , n, Li = Li−1(ti) and one of the following holds: either
1. ∂ti ∈ Li−1; we say ti is an integral (of an element of Li−1); or
2. ti 6= 0 and (∂ti) /ti ∈ Li−1; in such case, ti is an exponential (of an integral of an element of Li−1); or
3. ti is algebraic over Li−1.